Incremental Rendering of Deformable Trimmed NURBS Surfaces Gary K. L. Cheung† [email protected] † Rynson W.H. Lau† ‡ [email protected] Department of Computer Engineering & Information Technology, City University of Hong Kong, Hong Kong ‡ Department of Computer Science, City University of Hong Kong, Hong Kong ABSTRACT produce a variety of shapes simply by manipulating their control points and weights. However, NURBS surfaces are seldom used in interactive applications that demand realtime rendering performance because of their high rendering cost. There have been a lot of work carried out to address this problem. Most of the methods developed are based on tessellation [1, 6, 8, 9, 10, 17]. This tessellation process subdivides the NURBS surfaces into polygons so that the hardware graphics accelerator, if present, may render the polygons in real-time. However, this process is computationally very expensive. As a NURBS surface is deforming, this process must be executed in each frame to reﬂect the change of the object shape. Since in many real-time applications such as computer games, we may want to have many deformable objects in the environment. Existing rendering methods would be diﬃcult to render these objects in real-time. Trimmed NURBS surfaces are often used to model smooth and complex objects. Unfortunately, most existing hardware graphics accelerators cannot render them directly. Although there are a lot of methods proposed to accelerate the rendering of such surfaces, majority of them are based on tessellation, which is developed primarily for handling non-deforming objects. For an object that may deform in run-time, such as clothing, facial expression, human and animal character, the tessellation process will need to be performed repeatedly while the object is deforming. However, as the tessellation process is very time consuming, interactive display of deforming objects is diﬃcult. This explains why deformable objects are rarely used in virtual reality applications. In this paper, we present a eﬃcient method for incremental rendering of deformable trimmed NURBS surfaces. This method can handle both trimmed surface deformation and trimming curve deformation. Experimental results show that our method performs signiﬁcantly faster than the method used in OpenGL. Earlier, we proposed an eﬃcient method for rendering deforming NURBS surfaces [13]. The method pre-computes a polygon model and a set of deformation coeﬃcients for each deformable NURBS surface. During run-time, it incrementally updates the pre-computed polygon model of each deforming surface and progressively reﬁnes the resolution of the model according to the change in the surface curvature. We have shown that this method is much more eﬃcient than existing methods. Recently, we have applied this method to develop a distributed virtual sculpting system [14, 15]. We have also extended the method to cover various types of deformable parametric free-form surfaces [12]. However, we have learnt from our experience that in order to represent real objects, such as human faces with eyes and mouths, many NURBS surfaces need to be used. This is because our method can only support regular NURBS surfaces. To represent an object with holes, we need to combine many regular NURBS surfaces to model a single object. To overcome this limitation, our objective of this project is to extend the method to support trimming [5], which is a technique to allow arbitrary regions of a NURBS surface to be cut out, resulting in a non-regular NURBS surface. Categories and Subject Descriptors I.3.5 [Computational Geometry and Object Modeling]: Curve, surface, solid, and object representations; I.3.5 [Methodology and Techniques]: Interaction techniques Keywords Deformable objects, NURBS surfaces, trimmed surfaces, realtime rendering 1. Frederick W.B. Li† [email protected] INTRODUCTION Deformable objects have been considered as important to virtual reality applications, as they may model clothing, facial expression, human and animal characters. In particular, Non-Uniform Rational B-Splines (NURBS) [4, 16] are often employed to represent such objects as they can be used to Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. VRST'03, October 1-3, 2003, Osaka JAPAN. Copyright 2003 ACM 1-58113-569-6/03/0010...$5.00 In this paper, we describe a method to extend our NURBS rendering method [13] to support trimming. The rest of the paper is outlined as follows. Section 2 provides a brief survey on related work. Section 3 describes how we handle the deformable trimmed NURBS surfaces. Section 4 presents our method for tessellating trimmed NURBS surfaces. Sec- 48 method, like most of the other methods, is not suitable for interactive rendering of trimmed NURBS surfaces. tion 5 describes how to incremental update trimming curves and trimmed NURBS surfaces as they deform. Section 6 demonstrates the performance of the method through some experiments. Finally, section 7 brieﬂy concludes the paper. 2. 3. HANDLING OF DEFORMABLE TRIMMED NURBS SURFACES RELATED WORK In our earlier work, we developed a technique for eﬃcient rendering of deformable NURBS surfaces [11, 13]. The basic idea of this method is to maintain two data structures of each surface, the surface model and a polygon model representing the surface model. As the surface deforms, the polygon model is not regenerated through tessellation. Instead, it is incrementally updated to represent the deforming surface. To handle trimming curves eﬃciently, we also model them in a similar way. Eﬃcient rendering of trimmed NURBS surfaces has been a challenging research area for decades. There are many methods proposed for tessellating trimmed NURBS surfaces into polygons for rendering. In particular, Shantz et al. [18] propose an adaptive forward diﬀerencing technique to evaluate the points on the surface incrementally to produce a polygon model for rendering. As the method may adjust the forward diﬀerencing step adaptively, it could optimize the number of surface points generated according to the surface curvature or other approximation criteria. Rockwood et al. [17] propose an alternative method to accelerate the tessellation process. It ﬁrst converts the surface to Bézier patches with knot insertion. For those trimmed patches, it further subdivides them into a list of uv-monotone patches. Each patch is then tessellated into polygons by the coving and tiling process. A variant of this method has been implemented in the OpenGL library. 3.1 Handling of the NURBS Surfaces The polygon model of a surface can be obtained by evaluating the surface equation with some discrete parametric values. If a control point is moved from Pi,j to P i,j with a → − displacement vector V = P i,j − Pi,j , the incremental diﬀerence between the two polygon models of the surface before and after the control point movement is: → − (P i,j − Pi,j )wi,j Ni,p (u)Nj,q (v) = αi,j V S(u, v) − S(u, v) = m n w N (u)N (v) t,q s=0 t=0 s,t s,p (1) where S(u, v) and S(u, v) are the polygon models of the surface before and after the control point movement, respectively. αi,j is called the deformation coeﬃcient deﬁned as follows: Abi-Ezzi et al. [1] tessellate trimmed NURBS surfaces in a way similar to [17], but it further minimizes the number of patches needed to be tessellated by culling out the invisible patches dynamically during run-time. Kumar et al. [9] improve the performance of the tessellation process by avoiding the operation of subdividing Bézier patches into uvmonotone patches. Instead, it directly tessellates the Bézier patches into polygons for rendering. However, to allow fast back-patch culling, it needs to compute pseudo normal surfaces for all Bézier patches. In [10], they enhanced their method in [9] by constructing super-surfaces on the Bézier patches to allow a further reduction in the number of polygons of the resulting polygon model. wi,j Ni,p (u)Nj,q (v) αi,j (u, v) = m n s=0 t=0 ws,t Ns,p (u)Nt,q (v) (2) αi,j is a constant for each particular pair of (u, v). Hence, if the resolution of the polygon model does not change during the deformation, we may precompute the deformation coeﬃcients and update the polygon model incrementally as shown in Equation (1). Unfortunately, all the methods mentioned above cannot handle deforming trimmed NURBS surfaces eﬃciently. Regardless of the high computational cost of the tessellation process, it has to be performed repeatedly in every frame as the surfaces are deforming. On the other hand, if the trimming curve is undergoing deformation, methods adopting the uvmonotone approach will even need to re-generate a new set of uv-monotone patches before the tessellation process. However, when a surface deforms, its curvature is likely changed and we need to reﬁne the resolution of the polygon model and to compute new deformation coeﬃcients incrementally according to the change in the surface curvature. A NURBS surface is ﬁrst converted into a set of Bézier patches using knot insertion [2]. Each Bézier patch is then subdivided into a polygon model, which is maintained in a quadtree hierarchy, by applying the de Casteljau subdivision formula [3] to the Bernstein polynomials in both u and v directions. We refer to this as the polygon hierarchy. For example, in u, we have: Recently, Kahlesz et al. [8] developed an adaptive tessellation method. It recursively subdivides a surface into a quad-tree hierarchy according to some approximation criteria. If a leaf quad-tree node is found to be untrimmed, it will be selected as an output polygon for the resulting polygon model. Otherwise, the node is further subdivided for further processing or be tessellated by constrained delaunay triangulation to generate the output polygons. This method was further enhanced by building a seam graph structure on the tessellated trimmed NURBS surfaces for multi- resolution modeling [6]. A seam graph structure is actually a progressive mesh [7] like structure. It allows the application to select an appropriate resolution of the tessellated polygon model for rendering. However, the construction of the seam graph structure is itself an expensive process. Hence, this w wir−1 r−1 P (u) + u i+1 P r−1 (u) wir i wir i+1 r−1 Pir (u) = (1 − u) (3) r−1 (u) and r = 1, . . . , n, where wir (u) = (1 − u)wir−1 (u) + uwi+1 T i = 0, . . . , n − r. [ w P w ] are the homogeneous Bézier i i i points with Pi ∈ R3 , wi are the weights, and n is the degree of the surface. The v direction has similar recursion. The diﬀerence of Equation (3) before and after the deformation can be simpliﬁed to get a de Casteljau formula as 49 node region is trimmed out by the trimming curve as shown in ﬁgures 1(a) and 1(b), respectively. The former represents a convex trimmed node and can be tessellated by a simple triangulation algorithm. The latter may represent either a monotonic chain trimmed node or a complex trimmed node. follows: αri (u) = (1 − u)αr−1 (u) + uαr−1 i i+1 (u) (4) for r = 1, . . . , n, i = 0, . . . , n−r. Equation (4) indicates that the deformation coeﬃcients can be generated incrementally by the de Casteljau subdivision formula. Hence, if the resolution of the polygon model needs to be increased, the new deformation coeﬃcients can be calculated from adjacent deformation coeﬃcients stored at existing vertices using the de Casteljau formula. To achieve a better performance, we implemented this based on the Horner’s formula, of average complexity O(n) as opposed to O(n2 ) when based on the de Casteljau’s formula. 3.2 Handling of the Trimming Loops A trimmed NURBS surface is deﬁned by a set of trimming loops together with the NURBS surface itself. Each trimming loop consists of a set of NURBS curves, which are deﬁned over the parametric space of the NURBS surface. A NURBS curve C(t) may be deﬁned as: C(t) = n βi (t)Pi (a) (b) Figure 1: Note trimming: (a) exterior node region is trimmed out, and (b) interior node region is trimmed out. (5) i=0 where Pi denotes the control points. βi (t), similar to the case of NURBS surfaces, represents the set of deformation coeﬃcients of the NURBS curve. It is deﬁned as: wi Ni,p (t) βi (t) = m s=0 ws Ns,p (t) 4.2 Monotonic Chain Trimmed Nodes A polyline segment is a monotonic chain with respect to axis L if the polylines of the inscribed trimming curve segment have at most 2 intersections to any L perpendicular to L. It is similar to the monotone deﬁnition but a monotone refers to a polygon while a monotonic chain refers to the polylines. In our method, the monotonic chain is respected to the u and the v axes in the parametric space as uv-monotonic polylines. A monotonic chain trimmed node is then a combination of the uv-monotonic polylines and the corners of the trimmed node as shown in ﬁgure 2. By tessellating these monotonic regions as a whole, we can both reduce the number of node subdivisions and minimize the number of resulting polygons. As shown in ﬁgure 2(a), such case exists when there are u-monotonic and v-monotonic polylines inscribed in a node, in which the monotonic polylines are connected at their maxima, umax and vmax , and minima, umin and vmin , as shown in ﬁgure 2(b). In other words, there are four uv-monotonic polylines in the node, which have the following properties: (6) To trim a NURBS surface, we subdivide the NURBS curves against the polygon hierarchy of the NURBS surface to form polylines. By comparing the intersection points between the polylines and the polygon hierarchy in the parametric space, we may obtain a list of polylines for each quadtree node in the polygon hierarchy that are inscribed in the node. 4. TESSELLATION OF TRIMMED NODES To render a trimmed NURBS surface, according to the current viewing parameters, we select the appropriate quadtree nodes in the polygon hierarchy and tessellate them based on their types. We classify all the nodes into three types: visible non-trimmed nodes, invisible non-trimmed nodes and trimmed nodes. For a visible non-trimmed node, we just need to split it into 2 triangles along its diagonal. For an invisible non-trimmed node, as it is completely trimmed out by some trimming loop, we would just ignore it and would not render it. For a trimmed node, we further classify it into one of the three types: convex trimmed node, monotonic chain trimmed node and complex trimmed node. They are described in the following subsections. v max M upper-left M upper-right L1 u min A convex trimmed node is a node with a convex trimmed region. To classify such a node, we make use of the strong convex hull property of the NURBS curve deﬁnition. We perform an angle test on the control points of the curve segment to verify the convexity of the trimmed region in the node. Once such a node is identiﬁed, there are two possible cases as shown in Figure 1. Either the exterior or the interior L3 L4 M lower-left v min 4.1 Convex Trimmed Nodes L2 M lower-right u max (a) (b) Figure 2: Handling of a monotonic chain trimmed node: (a) determining maxima and minima, and (b) partitioning of the node into four regions according to the maxima and minima. 50 In u direction: determinant of the 3 vertices is calculated as follows: ui − ulef t (∆x + ui ) − ulef t vi − vupper (∆y + vi ) − vupper ∀x{x : (umin ≤ ux ≤ ux+1 ≤ umax ) ∧ (ux , ux+1 ) ∈ Lupper } ∀x{x : (umax ≥ ux ≥ ux+1 ≥ umin ) ∧ (ux , ux+1 ) ∈ Llower } = In v direction: = ∀y{y : (vmin ≤ vy ≤ vy+1 ≤ vmax ) ∧ (vy , vy+1 ) ∈ Llef t } [∆y(ui − ulef t ) + (ui − ulef t )(vi − vupper )] − [∆x(vi − vupper ) + (ui − ulef t )(vi − vupper )] ∆y(ui − ulef t ) − ∆x(vi − vupper ) (7) As ulef t ≤ umin and ∆y ≥ 0, the ﬁrst part, ∆y(ui − ulef t ), of equation 7 must be positive. In addition, as vupper ≥ vmin and ∆x ≥ 0, the second part, −∆x(vi − vupper ), of equation 7 must also be positive. Therefore, the result of equation 7 must always be positive. ∀y{y : (vmax ≥ vy ≥ vy+1 ≥ vmin ) ∧ (vy , vy+1 ) ∈ Lright } where Lupper and Llower denote the upper uv-monotonic polylines (L1 ∪ L2 ) and the lower uv-monotonic polylines (L3 ∪ L4 ), respectively. These polylines are partitioned by {umax , umin }. Llef t and Lright denote the left uv-monotonic polylines (L1 ∪ L4 ) and the right uv-monotonic polylines (L2 ∪ L3 ), respectively. These polylines are partitioned by {vmax , vmin }. By combining the 4 uv-monotonic regions together, the result of the tessellation process may become as shown in ﬁgure 4. Since this tessellation process requires to identify the 2 pairs of maxima and minima, the complexity is O(n) bounded. In addition, the process traverses each vertex of the polylines at most once, which is also O(n) bounded. However, if a node is identiﬁed as a non-monotonic chain trimmed node, i.e., it is a complex trimmed node, a further subdivision is required. All four uv-monotonic regions share the same properties, except that they have diﬀerent orientations. To show how we tessellate the node, we consider the upper-left uv-monotonic region, Mupper−lef t , as shown in ﬁgure 3. Since a uv-monotonic region is convex in nature, a simplest way to tessellate it is by joining each vertex of the monotonic polylines within the region to the nearest corner point of the region. As an example, the nearest corner point of Mupper−lef t as shown in ﬁgure 3 is Pupper−lef t , which has the coordinate (ulef t , vupper ). Since the uv-monotonic polylines are inscribed in the node, we can have u lef t ≤ umin and vupper ≥ vmin . The tessellation is done by adding lines from corner point Pupper−lef t to each vertex Pi on the uv-monotonic polylines of Mupper−lef t . P upper-left(u left,v upper) Figure 4: The resultant tessellation of a monotonic chain trimmed node. x P i+1(u i+1,v i+1) 4.3 Complex Trimmed Nodes When a node is identiﬁed as neither a convex trimmed node nor a monotonic chain trimmed node, it is considered as a complex trimmed node. Normally, if a node contains a highly irregular trimming curve, it is most likely a complex trimmed node. To handle this kind of nodes, we need to further subdivide each of them into child nodes. This may likely also involve subdividing the trimming curve to individual child nodes. Hence, this subdivision process essentially reduces the irregularity of the trimming curve and allows more and more subnodes to be classiﬁed as convex trimmed nodes or monotonic chain trimmed nodes for triangulation. On average, each child node contains about one-fourth of the original trimming curve. Such a reduction in irregularity is very eﬀective. According to our experiments, more than 80% of nodes in a surface would be subdivided into convex trimmed nodes or monotonic chain trimmed nodes. Only 20% of the nodes need to be further subdivided. From our experience, most nodes require only 1 or 2 levels of subdivision to partition a complex trimmed node into convex trimmed nodes and/or monotonic chain trimmed nodes. y P i(u i,v i) Figure 3: Tessellation of the upper-left uv-monotonic region. To show that this tessellation process would work correctly, i.e., the lines added would not cross each other, we consider two consecutive sample points of a trimming curve segment, i.e., the two end points of a polyline. Refer to ﬁgure 3 as an example. The two end points are Pi and Pi+1 , with Pi+1 being the next point of Pi . The two points form a triangle with Pupper−lef t and the coordinates of the triangle are (ulef t , vupper ), (ui , vi ) and (ui+1 , vi+1 ). If the orientation of these 3 vertices is always turning left, their determinant should then be positive. To evaluate the determinant, we re-express Pi+1 as (∆x + ui , ∆y + vi ), where (∆x, ∆y) is the oﬀset from Pi to Pi+1 . According to the monotone property, ∆x and ∆y will always be positive in Mupper−lef t . The 51 5. to decrease the resolution of C. By observation, an update to C(t 3 ) will aﬀect the polylines between C(t1 ) to C(t5 ). In fact, C(t1 ) and C(t5 ) are the previous vertex and the next vertex to C(t3 ), respectively. Hence, C(t1 ) and C(t5 ) may be considered as the updated range of C(t 3 ). Generally speaking, whether a resolution reﬁnement process involves inserting or deleting vertices, the update range always falls between (max{t : t < ti }, ti ) and (ti , min{t : ti < t}). OBJECT DEFORMATION One of the critical developments in our research is that our method supports real-time deformation of both the trimming curves and the trimmed NURBS surfaces. We note that the deformation of a trimming curve does not aﬀect the topology or the shape of the trimmed NURBS surface. On the other hand, the deformation of the trimmed NURBS surface will only aﬀect the shape of the surface itself; it does not aﬀect the shape of the trimming curve. As such, both types of deformation are only loosely related to each other, and we can handle each of them separately. C(t 2) 5.1 Trimming Curve Deformation C(t 3) C(t 4) C(t 5) C(t 1) Usually, when a trimming curve is being deformed, only part of the NURBS surface is aﬀected. It will be expensive to perform the retriangulation of the trimming curve against the polygon hierarchy of the NURBS surface in every frame while the trimming curve is deforming. According to the local modiﬁcation property of the NURBS curves, any deformation driven by moving a control point Pi of a NURBS curve will only aﬀect the curve segment within the parametric range [ti , ti+p ), where p is the order of the trimming curve. Hence, we may limit the update operation to within this trimming curve segment deﬁned by ti and ti+p . To allow an eﬃcient update of the corresponding polylines of the curve segment, we may simply check each of the polylines to see if its two end points satisfy the following condition: C(t 1) C(t 2) C(t 3) C(t 5) C(t 1) C(t 1) C(t 5) C(t 5) C(t 3) (ti ≤ tstart ∧ tend ≤ ti+p ) C(t 4) C(t 3) C(t 2) C(t 4) Inserting Child Nodes C(t 2) C(t 4) Deleting Child Nodes (a) (b) Figure 5: Resolution refinement of a trimming curve segment: (a) resolution increase, and (b) resolution decrease. Only those polylines satisfying this condition are aﬀected by the deformation and hence need to be updated. Once we have identiﬁed the aﬀected polylines, we would update the parametric positions of their vertices. We then perform resolution reﬁnement and compute the new sets of deformation coeﬃcients for the newly inserted vertices. The updated polylines are then remapped to the NURBS surface for retessellation. Note that the curvature change of the trimming curve segment due to the deformation of the NURBS surface is usually very small. Hence, only a very small number of vertices may need to be inserted or deleted from the polylines in order to maintain a good polygonal representation of the trimmed surface. Figure 6 shows a trimmed NURBS surface before and after deformation. We can see that both the resolution of the surface itself and the resolution of the trimming curve are reﬁned according to the change of the surface curvature. 5.2 Trimmed Surface Deformation When a trimmed NURBS surface deforms, the curvature of the surface may be aﬀected and the curvature of the trimming curves within the deformed region of the surface may also be aﬀected. We handle this in a way somewhat similar to how we handle the deformation of the trimming curve. However, the scope of the update is relatively smaller. First, we update the aﬀected region of the NURBS surface. Second, we update the vertex positions of the aﬀected polylines on the NURBS surface. Third, we perform resolution reﬁnement on the aﬀected trimming curve segment. Finally, we retessellate the resulting polylines with the corresponding nodes. Figure 5 illustrates two examples of resolution reﬁnement of a trimming curve segment C with reference to a node in the polygon hierarchy, where C(ti ) denotes a vertex of the polylines representing C. In ﬁgure 5(a), we assume that the polylines can no longer approximate C due to the increase in the surface curvature. Hence, we need to increase the resolution of C by inserting vertices C(t2 ) and C(t4 ) to the polylines. On the other hand, if the resolution of C is found to be too high due to the decrease in the surface curvature as shown in ﬁgure 5(b), we may delete vertices C(t2 ) and C(t4 ) (a) (b) Figure 6: Resolution refinement of a trimmed NURBS surface: (a) before deformation, and (b) after deformation. 52 Time (Sec) Time (Sec) Trimming Curve Deformation Only 0.120 0.120 0.100 0.100 0.080 0.080 0.060 0.060 0.040 0.040 OpenGL Our Methods 0.020 0.000 0 100 200 300 Frame Number Surface Deformation Only OpenGL Our Methods 0.020 0.000 400 500 0 100 200 300 Frame Number 400 500 Figure 7: Performance comparison between our method and the method used by OpenGL. Time (Sec) Time (Sec) Trimming Curve Deformation Only 0.050 Surface Deformation Only 0.050 Rendering Coordinate Update Tessellation 0.040 Rendering Coordinate Update Tessellation 0.040 0.030 0.030 0.020 0.020 0.010 0.010 0.000 0.000 0 100 200 300 Frame Number 400 500 0 100 200 300 Frame Number 400 500 Figure 8: Computational costs of individual operations. 6. RESULTS AND DISCUSSIONS If we compare trimming curve deformation with trimmed surface deformation as shown in ﬁgure 7, we can see that the performance of trimming curve deformation is relatively ﬂuctuating. This is because whenever a trimming curve is deformed, we need to perform the retessellation process. The eﬃciency of this retessellation process is mainly aﬀected by the curvature of the trimming curve, since a smoother trimming curve will generate less polylines. For the randomly deforming trimming curves, their curvatures are expected to vary signiﬁcantly. As a result, the computation time would be unstable. In contrast, the surface deformation does not generate a massive update of trimmed surface patches and hence, the computation time is relatively stable. We have implemented the new method in C++ and OpenGL. All the experiments presented here have been performed on a Pentium 4 2.0GHz machine with 256 MB RAM. The deformation events were triggered by randomly moving either 40% of the trimming curve control points, or 5% of the surface control points in each frame. Figure 9 shows a human face model that we used to test the proposed rendering method. The model is constructed by a single NURBS surface containing 837 control points and 9 separate trimming curves. The ﬁgure shows the model before and after deformation, in shaded and wireframe rendering. Figure 10 shows another test model of a classical teapot after deformation. Figure 8 shows the performance of individual operations used in our method. Tessellation refers to the classiﬁcation and triangulation of all the trimmed nodes. Coordinate update refers to the incremental updating of the trimming curves and the trimmed surface. Rendering refers to the time taken to render all the polygons by the OpenGL engine in each frame. We can see that the tessellation process only needs to be executed occasionally in the surface deformation case. For trimming curve deformation, as the surface coordinates of the polylines cannot be calculated with the deformation coeﬃcients, they should be recomputed from the coordinates of the incrementally updated trimming curve. Hence, this process is in general more computationally intensive for trimming curve deformation than for trimmed To demonstrate the performance of our method, we have compared it with the method used by OpenGL, i.e., the variant of Rockwood’s method [17], using the human face model shown in ﬁgure 9. Figure 7 shows the rendering times of the two methods, we can see that our method in each consecutive frame is roughly 2.5 and 3 times faster than the OpenGL method for trimming curve deformation and for trimmed surface deformation, respectively. The reason for this is that as we deform the trimmed surface and the trimming curves continuously, a complete evaluation is required in each frame for the OpenGL method. However, our method only needs to perform an incremental update to the aﬀected region and is thus more eﬃcient. 53 [9] S. Kumar, D. Manocha, and A. Lastra. Interactive Display of Large-Scale NURBS Models. In Proc. of Symposium on Interactive 3D Graphics, pages 51–58, 1995. surface deformation. When a trimming curve deforms, we need to perform re-evaluation on the curve to generate the updated polylines representing the deformed curve. However, in practice, this process may not signiﬁcantly degrade the overall rendering performance as the region of the deformation, and hence, the eﬀort spent on the update, is usually relatively small compared with the deformation of the whole surface. 7. [10] S. Kumar, D. Manocha, H. Zhang, and K. Hoﬀ. Accelerated Walkthrough of Large Spline Models. In Proc. of Symposium on Interactive 3D Graphics, pages 91–101, 1997. [11] F. Li and R. Lau. Incremental Polygonization of Deforming NURBS Surfaces. Journal of Graphics Tools, 4(4):37–50, 1999. CONCLUSION In this paper, we have introduced an interactive rendering method for deformable trimmed NURBS surfaces. In order to eﬃciently update the trimmed NURBS surface and the trimming curves as the surface and/or the trimming curves are deforming, we propose a regional update mechanism and an eﬃcient method for dynamically tessellating the NURBS surface with the trimming curves. We have shown that the new method for rendering deformable trimmed NURBS surfaces is more eﬃcient than the method used in OpenGL during the curve/surface deformation. 8. [12] F. Li and R. Lau. Real-Time Rendering of Deformable Parametric Free-Form Surfaces. In Proc. of ACM VRST, pages 131–138, December 1999. [13] F. Li, R. Lau, and M. Green. Interactive Rendering of Deforming NURBS Surfaces. In Proc. of Eurographics ’97, pages 47–56, September 1997. [14] F. Li, R. Lau, and F. Ng. Collaborative Distributed Virtual Sculpting. In Proc. of of IEEE VR’01, pages 217–224, March 2001. ACKNOWLEDGMENTS [15] F. Li, R. Lau, and F. Ng. VSculpt: A Distributed Virtual Sculpting Environment for Collaborative Design. IEEE Trans. on Multimedia (to appear), 2003. The work described in this paper was partially supported by two SRG grants (Project Numbers: 7001391 and 7001465) and a DAG grant (Project Number: 7100264), all from City University of Hong Kong. 9. [16] L. Piegl and W. Tiller. 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Fast and Memory Eﬃcient View-Dependent Trimmed NURBS Rendering. In Proc. of of Paciﬁc Graphics ’02, pages 204–213, June 2002. [7] H. Hoppe. Progressive Meshes. In Proc. of ACM SIGGRAPH ’96, pages 99–108, August 1996. [8] F. Kahlesz, A. Balázs, and R. Klein. Multiresolution Rendering By Sewing Trimmed NURBS Surfaces. In Proc. of ACM Symposium on Solid Modeling and Applications, pages 281–288, June 2002. 54 Figure 9: A human face modeled by a trimmed NURBS surface: (upper) shaded and wireframe before deformation, and (lower) shaded and wireframe after deformation on the eyes, mouth and face. Figure 10: A teapot modeled by a trimmed NURBS surface: shaded and wireframe before deformation. 55

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